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Records in 5-almost prime gaps ordered by merit.
2

%I #12 Jan 22 2025 02:31:40

%S 16,24,28,42,56,70

%N Records in 5-almost prime gaps ordered by merit.

%C Next term, if it exists, is associated with indices above 100000 in A114405 and A014614. - _R. J. Mathar_, May 10 2007

%F a(n) = records in A114405(n)/log_10(A014614(n)) = records in (A014614(n+1) - A014614(n))/log_10(A014614(n)).

%e Records defined in terms of A114405 and A014614:

%e n A114405(n) A114405(n)/log_10(A014614(n))

%e = ========== =============================

%e 1 16 16/log_10(32) = 10.6301699

%e 2 24 24/log_10(48) = 14.2751673

%e 3 8 8/log_10(72) = 4.30725248

%e 4 28 28/log_10(80) = 14.7129144

%e 5 4 4/log_10(108) = 1.96712564

%e 6 8 8/log_10(112) = 3.90392819

%e 7 42 42/log_10(120) = 20.2002592

%e 8 6 6/log_10(168) = 2.69625443

%e ...

%e 22 56 56/log_10(312) = 22.4524976

%p A014614 := proc(nmax) local a,i; a := [] ; i := 1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 5 then a := [op(a),i] ; fi ; i := i+1 ; end: a ; end: A114405 := proc(a014614) local a,i; a := [] ; for i from 2 to nops(a014614) do a := [op(a), op(i,a014614)-op(i-1,a014614)] ; od ; a ; end: a014614 := A014614(100000) : a114405 := A114405(a014614) : Digits := 30 : rec := -1 : for i from 1 to nops(a114405) do if evalf(a114405[i]/log(a014614[i])) > rec then printf("%d, ",a114405[i]) ; rec := evalf(a114405[i]/log(a014614[i])) ; fi ; od ; # _R. J. Mathar_, May 10 2007

%Y Cf. A014614, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Nov 25 2005

%E a(6) from _R. J. Mathar_, May 10 2007